Introduction
In the field of economics, business commerce and management forecasting is often required for several reasons.
Prices of commodities, agricultural production, mineral production, national income, prices of shares, number of passenger travelling from a certain place, revenue collection, population, volume of import and export, volume of foreign exchange, electricity consumption of a city are the examples where the changes take place every now and then. Thus prices, production, consumption are the main areas where the changes are bound to occur with respect to time. In the examples cited, above reliable forecasting needs the knowledge of the reasons behind the changes, the time epochs of changes and the magnitude of changes. The techniques and tools of forecasting are included in this chapter titled 'Time series'. Earlier, we have studied some forecasting tools such as regression lines, fitting and curves, interpolation etc. However, those tools are useful for longterm, having steady and smooth curve. In reality we rearely see a smooth curve; how to adjust for these temporary fluctuations is studied in this chapter. The examples given above in the field of commerce, economics are called as 'Time series'. The precise definition is discussed subsequently. Number of definitions of time series are available. Some of them are given below.
Meaning of Time Series
Definition: Time series is a series of statistical observations arranged in chronological order.
The above definition is due to Morris Hamburg. The observations in the chronological order means in the order of occurrence, taken at a regular successive intervals or points of time. The time intervals may be years, months, weeks, days, minutes and seconds also in some cases.
Following are some examples of time series.
1. Daily price of gold.
2.Weekly sales of departmental store.
3.Monthly deposits in a certain bank.
4. Yearly production of food grains in a certain country.
5.Daily record of maximum temperature in a city.
6.Hourly bacterial count in certain culture at laboratory.
7.Population of country at census years.
According to Spiegel, mathematically a time series is defined by the values Y1, Y2,..., Yn, ... of the variable Y at times t1, t2, tn,. Thus, time series is a function of time i.e. Y = F (t). In other words, in time series time plays the role of an independent variable and Y (t) is dependent variable. We denote time series by Y (t) or Yt. In the form of function the time series may be written as follows:
t:
Yt:
t1 Y1
tz Y2
...
...
tn Yn
The time points t1, t2, t3, ... are equidistant. The analysis of time series
is important from many aspects. We plot t on X-axis and Yt on Y-axis. Consider, the example of population of India in census years.
Year (t) 1901 1911 1921 1931 1941 Population 23.84 25.21 25.13 27.9 31.87 (Crores) Yt
Year (t) Population (Crores) Yt
1951 1961 1971 1981 1991 2001 2011 36.11 43.92 54.82 63.33 84.39 102.70 121.12
We observe that the population is measured during census years with intervals of ten years. Hence, the figures are arranged chronologically, that is in order of time. If they are not ordered according to time, then they would not be able to provide information about the pattern of variable. If you carefully observe the above time series, you will notice that the population has an increasing trend. Now have a look at the following graph.
The following things are worth noticing.
(i) India's population is steadily increasing since 1921.
(ii) The rate of population increase accelerates further since 1951,
that is, after independence.
Prediction for year 2050 is 163 crores.
A time series is a summary of past information. Assuming that the 'past' serves a guide to the 'future', time series analysis helps in detecting the underlying patterns and projecting them into future.
Some authors describe the time series data represented graphically as historigram.
The nature of time series graph is usually not smooth and not monotonic, it is zig-zag or haphazard. The critical study reveals that these fluctuations are not totally haphazard, however some part is systematic and the only counter part is haphazard in nature.
Utility of Time Series Analysis
Time series analysis is of paramount interest in various disciplines such as economics, business, social sciences. Its uses are discussed. below:
(i) Past behaviour: It enables to describe the past behaviour of the variables. Time series analysis reveals the forces working behind the series such as technological and economical developments, changes in import, export policies.
(ii)Forecasting: Forecasting is one of the important use of analysis of time series. The forecasting in business plays important role in planning decision-making, inventory, scheduling of purchases and sales etc.
Isolating and measuring the effects of various components help the investigator to forecast the value of variable in future with fairly good reliability.
(iii) Comparison: Time series analysis facilitates the comparison between the two related time series. For example,
(a) Prices of gold and prices of shares,
(b) National income and cost of living indices. Comparison between actual and expected performance can be made comparison between two similar time series at two different places.
Components of Time Series
Time series values of Yt is composed of four factors or components viz. (1) Trend, (2) Seasonal variations, (3) Cyclical variations, (4) Irregular variations. These factors cause fluctuations in the values of Yt. In other words the fluctuations in time series are classified into the above four patterns or categories. A time series may have some or all the components present in it.
Trend or Secular Trend (T):
The trend is the smooth, regular long term movement in time series. It is the general tendency of data. The time series oscillates around trend. The trend may be to go upward to go downword or to remain stagnant.
For example, the yearly population, yearly agricultural production, prices etc. show an upward trend. On the other hand cost of electronic goods, number of illiterates, yearly birth rates etc. show downward trend. Yearly rainfall, daily temperature atmospheric pressure at a certain place, monthly electricity consumption of a family are the examples of constant trend. The trend may be linear or non-linear in nature.
Remarks:
(i) Trend is also called as secular trend. The word secular is derived from the Latin word saeculum which means generation or age.
(ii) Trend is due to the reasons of following nature, changes in population, technological developments, changes in economy, changes in habits and tastes of people.
(iii)Trend is a long period movement however, the period cannot be precisely defined. For example, regarding price of gold, agricultural production the long period cannot be few weeks or 2, 3 years. may be observed over a period of 10 to 15 years even more than that. Long term period may change from series to series.
(iv) Trend is mostly monotonic although original time series s not. Trend-
Trend
Ay
Time series.
Time series
(v)
Time (a) Linear trend
(b) Non-linear trend Fig. 4.2
Time
In linear trend the values of Y can be approximated by a straight line. This indicates that the rate at 'which the time series values increase or decrease is constant. On the other hand, in non-linear trend, the growth rate is different over different sectors of time. Linear trend is commonly used in business and economics. Non-linear trend is used in the study of population birth rates etc.
(v)Apart from the long-term growth component, there are some short-term periodic rhythmic variations. These variations disturb the smoothness and monotony.
(vi)Trend is useful for two reasons:
(a) It facilitates the comparison of two time series.
(b)It helps to extrapolate.
Seasonal Variations (s):
Seasonal variations are the fluctuations in a time series which repeat regularly every year or after some specific period of time.
For example, sales of umbrellas and raincoats is the highest in rainy season; sales of wollen garments attains its peak in winter, sales of luxury items and jewellary is high during festivals. Even the bank deposits and bank clearings are affected by seasonal swings. Here, the 'seasons' may be taken as weeks. Similarly, traffic maximum during rush hours; so 'seasons' are hours in this case. The "seasons' may be seconds in bacterial population growth.
The seasonal variations may be either due to natural forces such as festivals climatic conditions or due to customs, fashions or habits of the people. These factors operate in a regular and periodic manner where the period of recurrence is generally one year. Following graph shows seasonal variations in the sales of umbrellas (Y) from a store.
Im
Year x
Fig. 4.3
Note that the sales attain maximum in the month of June every year. Remark: The amplitude may differ from cycle-to-cycle.
The study of seasonal variations is of prime importance in many time series. Particularly, when the trend is stagnant, seasonal variations become predominant component.
Seasonal variations are extremely useful in marketing and business field in many ways. For example, during summer sales of fans, coolers, refrigerator, cold-drinks, ice-creams increase considerably and reach the peak. Business man has to take care of inventory for seasonal peaks, he has to employ adequate number of salesmen, he has to schedule purchases and sales, he has to arrange for clearance sale, he has to advertise, and give discount on prices for off seasons, he has to arrange for additional finances during seasons. Similarly, bank managers has to arrange for proper cash flow during the beginning of month, festivals.
Cyclical variations (c) :
Cyclical variations in a time series are the fluctuations which repeat over a time period of more than one year. The cyclical variations may not be necessarily uniformly periodic. The amplitude of variation also changes from cycle to cycle. That is, the ups and downs may occur at different